Sliding Bifurcation Analysis and Global Dynamics for a Filippov Predator-prey System

Volume 9, Issue 6, pp 3948--3961 http://dx.doi.org/10.22436/jnsa.009.06.42

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Authors

Xuewen Tan - College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, P. R. China. - School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650500, P. R. China. Wenjie Qin - College of Science, China Three Gorges University, Yichang 443002, P. R. China. Xinzhi Liu - Department of Applied Mathematics, University of Waterloo, Ontario N2L 3G1, Canada. Jin Yang - Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, P. R. China. Shaoping Jiang - School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650500, P. R. China.

Abstract

This paper studies a Filippov predator-prey system, where chemical control strategies are proposed and analyzed. Initially, the exact sliding segment and its domains are addressed. Then the existence and stability of the regular, virtual, pseudo-equilibria and tangent points are discussed. It shows that two regular equilibria and a pseudo-equilibrium can coexist. By employing theoretical and numerical techniques several kinds of bifurcations are investigated, such as sliding bifurcations related to the boundary node (focus) bifurcations, touching bifurcations, sliding crossing bifurcation and buckling bifurcations (or sliding switching). Furthermore, it makes comparison of the obtained results with previous studies for the Filippov predator-prey system without control strategies. Some biological implications of our results with respect to pest control are also given.

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Keywords

• Filippov predator-prey system
• control strategy
• economic threshold
• sliding bifurcation analysis.

MSC

•  34H20
•  34K20

References

• [1] S. M. Baer, B. W. Kooi, Y. A. Kuznetsov, H. R. Thieme, Multiparametric bifurcation analysis of a basic two-stage population model , SIAM J. Appl. Math., 66 (2006), 1339-1365.


• [2] A. A. Berryman, B. A. Hawkins, The refuge as an integrating concept in ecology and evolution, Oikos, 115 (2006), 192-196.


• [3] B. Brogliato, Nonsmooth Mechanics, Springer-Verlag, London (1999)


• [4] C. A. Buzzi, T. D. Carvalho, P. R. da Silva , Canard cycles and Poincar index of non-smooth vector fields on the plane, J. Dyn. Control Syst., 2 (2010), 173-193.


• [5] C. A. Buzzi, P. R. da Silva, M. A. Teixeira, A singular approach to discontinuous vector fields on the plane, J. Differential Equations, 231 (2006), 633-655.


• [6] L. S. Chen, Z. J. Jing, Existence and uniqueness of limit cycles for the diferential equations of predator-prey interaction, Chinese Sci. Bull., 29 (1984), 521-523.


• [7] M. I. da Silveira Costa , Harvesting induced uctuations: insights from a threshold management policy, Math. Biosci., 205 (2007), 77-82.


• [8] M. I. da Silveira Costa, E. Kaszkurewicz, A. Bhaya, L. Hsu, Achieving global convergence to an equilibrium population in predator-prey systems by the use of a discontinuous harvesting policy, Ecol. Model., 128 (2000), 89-99.


• [9] M. I. da Silveira Costa, M. E. M. Meza, Application of a threshold policy in the management of multispecies fisheries and predator culling, Math. Med. Biol., 23 (2006), 63-75.


• [10] M. D. de Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. B. Nordmark, G. Olivar Tost, P. T. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Rev., 50 (2008), 629-701.


• [11] M. D. de Bernardo, P. Kowalczyk, A. Nordmark, Bifurcations of dynamical systems with sliding: derivation of normal-form mappings, Phys. D, 170 (2002), 175-205.


• [12] F. Dercole, A. Gragnani, S. Rinaldi , Bifurcation analysis of piecewise smooth ecological models, Theoret. Popul. Biol., 72 (2007), 197-213.


• [13] S. H. Doole, S. J. Hogan, A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation, Dynam. Stability Systems, 11 (1996), 19-47.


• [14] A. F. Filippov , Differential Equations with Discontinuous Righthand Sides, Translated from the Russian. Math- ematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht (1988)


• [15] G. F. Gause, The struggle for existence, Soil Science, (1934)


• [16] G. F. Gause, N. P. Smaragdova, A. A. Witt, Further studies of interaction between predators and prey, J. Animal Ecol., 5 (1936), 1-18.


• [17] V. Krivan , On the Gause predator-prey model with a refuge: a fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.


• [18] Y. Kuang, H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci., 88 (1988), 67-84.


• [19] Y. A. Kuznetsov, S. Rinaldi, A. Gragnani , One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.


• [20] M. E. M. Meza, A. Bhaya, E. Kaszkurewicz, M. I. da Silveira Costa, Threshold policies control for predatorprey systems using a control Liapunov function approach, Theoret. Popul. Biol., 67 (2005), 273-284.


• [21] W. Qin, S. Tang, The selection pressures induced non-smooth infectious disease model and bifurcation analysis, Chaos Solitons Fractals, 69 (2014), 160-171.


• [22] W. Qin, S. Tang, C. Xiang, Y. Yang , Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure, Appl. Math. Comput., 283 (2016), 339-354


• [23] S. Tang, R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257-292.


• [24] S. Tang, J. Liang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge, Nonlinear Anal., 76 (2013), 165-180.


• [25] S. Tang, J. Liang, Y. Xiao, R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J. Appl. Math., 72 (2012), 1061-1080.


• [26] S. Tang, Y. Xiao, L. Chen, R. A. Cheke, Integrated pest management models and their dynamical behaviour, Bull. Math. Biol., 67 (2005), 115-135.


• [27] S. Tang, Y. Xiao, N. Wang, H. Wu, Piecewise HIV virus dynamic model with CD4+ T cell count-guided therapy, J. Theoret. Biol., 308 (2012), 123-134.


• [28] V. I. Utkin, J. Guldner, J. X. Shi, Sliding mode control in electro-mechanical systems , second ed., CRC press (2009)


• [29] B. L. Van de Vrande, D. H. Van Campen, A. De Kraker, An approximate analysis of dry-friction-induced stick-slip vibrations by a smoothing procedure, Nonlinear Dyn., 19 (1999), 157-169.


• [30] Y. Xiao, H. Miao, S. Tang, H. Wu, Modeling antiretroviral drug responses for HIV-1 infected patients using differential equation models, Adv. Drug Deliv. Rev., 65 (2013), 940-953.


• [31] Y. Xiao, S. Tang, J. Wu, Media impact switching surface during an infectious disease outbreak, Scientific Rep., 5 (2015), 7838-7846.


• [32] J. Yang, S. Tang, R. A. Cheke, Global stability and sliding bifurcations of a non-smooth Gause predator-prey system, Appl. Math. Comput., 224 (2013), 9-20.


• [33] J. Yang, S. Tang, R. A. Cheke, Modelling the regulatory system for diabetes mellitus with a threshold window, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 478-491.